How to animate random turbulence on a mesh surface?
You can achieve this effect in many different ways. It will involve:
1) Particles / points that is wiggling around
2) Some kind of edge creations when points are closer together (KDtree).
3) Shaders to make it nice
4) Instancing

But I think the easiest and more elegant is using something like Animation Nodes Add-On or Sverchok.



Search: Plexus effect, Blender, kd-tree.

You might even found plug in that does this already.

Actually Animation Nodes has a template that does this kind of noise "plexus" effect or random points and network connection already. All you need to do is just make it pretty by adjusting the shader, transparency, etc.

And Sverchok has one node that generate KDTree edges... 

Enjoy!